I. 4.
RAINFALL ESTIMATION USING WEATHER RADAR AND GROUND STATIONS
Witold F. Krajewski
Weather radar is an attractive instrument for monitoring rainfall over large spatial domains. Its ability to provide detailed, in both time and space, information about precipitation patterns is unsurpassed by other operationally used sensors. However, since radar does not measure rainfall directly, radar-based rainfall estimation algorithms must be calibrated using direct observations. Such observations typically come from raingage networks. Raingage observations, although represent only point rainfall, are still considered as close to the true rainfall as we can get with the present day technologies. This is the main reason why raingage-based estimates of rainfall are used for validation of radar-based rainfall quantities.
In this paper we discuss this dual role of raingage observations of rainfall in calibration and validation of radar derived rainfall estimates. The use of raingage observations in radar calibration can be conceptualized as a three-stage process. The first is the parameter estimation of the basic relationship between radar-measured reflectivity and rainfall rate, i.e. Z-R relationship. Traditionally, the Z-R relationship is given in terms of two parameters a and b of a power type function, Z=aRb. The second stage involves adjustment of the mean field bias. Rainfall patterns, integrated in time to represent rainfall accumulations, are adjusted to raingage-based areal average of the corresponding rainfall. The adjustment should account for the sampling error of the typically sparse raingage network. The adjustment factor can be used in a predictive mode to adjust radar estimates in a time period following the adjustment and prior to arrival of the new raingage observations. The third stage may use raingage observations to locally adjust radar-rainfall patterns by optimal merging of the two sets of rainfall estimates according to some performance criterion, e.g. minimum error variance.
Not all operational radar installations utilize algorithms that include all three stages. The above design is probably better suited for real-time applications, such as the NEXRAD program in the United States (Hudlow et al., 1983; Ahnert et al., 1983) but similar conceptualization is also attractive for off-line studies of rainfall (Steiner et al., 1995; Ciach et al., 1995).
Validation is recognized as an important step in any application of radar-rainfall. As such applications may have widely varied requirements with respect to space-time resolution and accuracy, a general approach to validation should account for different factors that may become important at different scales. For example, validation of high spatial resolution rainfall has to deal with the problem of zero-rainfall intermittence, small-scale rainfall variability, and raingage measurement error. As the time integration increases, some of these problems become less pronounced and the validation is easier.
In the following we discuss different aspects of radar-rainfall calibration and validation. The discussion is illustrated with examples from various projects and applications. The scope of the paper is limited to conventional radar which measures only one parameters, i.e. radar reflectivity. Recommendations for future research close the paper.
Conversion of radar-measured quantity, i.e. reflectivity factor to a quantity of interest, i.e. rainfall rate, is a fundamental block of any radar-based rainfall estimation algorithm. This conversion can be accomplished using many different approaches, the most popular, however, is the use of power type function with two coefficients a and b. Relationship of this type should be regarded as empirical although strong theoretical justification exists for this choice. The justification is the fact, that both radar reflectivity and rainfall rate can be expressed as moments of drop size distribution within a radar sampling volume. This is where the first problem appears with using raingage observations for parameter estimation of Z-R relationship. Most rain gages do not measure rainfall rate but rather, rainfall accumulation. The radar sampling volume is often located at elevation as high as 1-2 km, with potentially significant time displacement (up to several minutes) which creates difficulties in time and space synchronization of the two measurements. Also, the radar sampling volume is much larger than the raingage sampling volume. These problems, when combined with extremely high space and time variability of rainfall, indicate that one should not expect high correlation between raingage observed rainfall and radar-estimated rainfall at short time scales.
Indeed, the high variability of drop size distribution alone, without the added radar measurement error and the sampling problems alluded to above, implies that Z-R relationship exists only in a statistical sense. Figure 1 shows some 5000 minutes of drop size distribution data obtained from Darwin, Australia (Short et al., 1990). When radar measurement process variability is added, the obtained comparisons of radar-estimated rainfall and raingage-observed 5-minute rainfall accumulations display correlation of only 0.5-0.6. For illustration, Figure 2 shows one month (January 1993) of collocated radar reflectivities and 5-minute raingage accumulation obtained from a C-band radar in Darwin, Australia.
The above problem is compounded by the difficulty in choosing radar observable to be converted to rainfall. Our common interest in rainfall on the ground suggest that only low antenna elevation angle data should be used. On the other hand, from the data quality control point of view, low elevation data are more likely contaminated by ground clutter, especially in anomalous propagation conditions. Also, the high vertical variability of radar reflectivity (Fabry et al., 1994) and our desire to mitigate the problems associated with time displacement of the compared radar-raingage samples, suggest that vertical integration of the radar signal should be considered. Such integration should be performed carefully to avoid contamination due to bright band. We will return to this point in the Validation section
The difficulties in calibrating the fundamental relationship between radar reflectivity and rainfall may be overcome as interest is shifted towards higher space and time scales. Time integration, in particular, should turn useful if the random component of the errors of the radar-based estimates is not strongly autocorrelated, and if there is a mechanism for removing systematic error (bias). Assuming that the first condition is true (on the basis that there is no strong evidence to the opposite), and considering that radar observations can be taken as often as 5 minutes, suggests that hourly radar-rainfall should be much better correlated with hourly raingage observations. Indeed, in the case of the radar data presented in Figure 2, time integration leads to correlation increased to about 0.7 at the hourly time scale. Does this represent radar precision in estimating rainfall? How can we determine radar-rainfall accuracy and precision? We will return to these questions later.
Fig. 1 - Rainfall rate and radar reflectivity calculated from drop size distribution data collected in Darwin, Australia.
Fig. 2 - Comparison of radar reflectivity and 5-minute raingage rainfall accumulation for 22 locations near Darwin, Australia.
Radar-rainfall estimation algorithms often involve much more than just a Z to R conversion. Thus, other elements of the algorithm can be parameterized and calibrated using raingage observations. Certain elements of the algorithm may even compensate for each other. For example, the multiplier in the power-type Z-R relationship, can be adjusted at the level of final products of the estimation process. In operational hydrology, such products are often hourly rainfall accumulation maps grided with resolution 2-4 km. In the following section we will examine principles of real-time adjustments of radar-based estimates.
The main idea behind real-time adjustment of radar-based estimates of rainfall accumulations is the need for frequent routine checks between rainfall reported from the existing raingage networks and the radar estimates. Despite all possible efforts in calibrating a given algorithm, including its Z-R relationship, the complexity of the radar instrumentation and measurement process, as well as the complexity and enormous variability of rainfall process, suggest discrepancies in radar-rainfall and raingage rainfall products. The sources of these discrepancies may be as simple as electronic miscalibration of the radar instrument, or a false Z-R relationship. More difficult to recognize sources of systematic(but time-varying) errors may exist due to geometric effects of the storms vs. the scanning strategy, and spatial intermittence of rainfall. Also, operational hydrologic models used for river flow forecasting are calibrated using the historical raingage databases and, if their performance is satisfactory, it is reasonable to request that on average radar estimates should not deviate to far from those based on gage observations.
On the other hand, we must also recognize that a typical raingage network is too sparse to accurately estimate areal mean of complex rainfall patterns. Considering that on average, hourly rainfall covers less than 10% of a given area, many storm will either go undetected or will be reported based on very few gages. Thus, it would be unwise to base our decision on adjusting radar-rainfall on such sparse reports subject to significant sampling errors. A good real-time bias adjustment scheme should account for these difficulties. Another desirable feature is time persistence. Although time constant of many convective storms may be as small as 30 minutes, rainfall patterns display certain degree of spatial organization that persist over certain time. Thus, if the existing bias displays similar temporal persistence, this fact can be used in predicting the necessary adjustment prior to receiving raingage reports. Once received, the raingage reports could be used in updating the adjustment values and a new predicted factor could be calculated.
Summarizing, the requirements for a real-time mean field bias adjustment should include the following elements: (1) the scheme should allow for real-time updating using new information; (2) sampling error due to limited raingage network density should be taken into account. There are several schemes described in the literature that satisfy these requirements (Hudlow et al., 1983; Lin and Krajewski, 1989; Smith and Krajewski, 1991; Seo et al., 1995). All of them deal with hourly rainfall and are based on the following definition of the mean field bias: it is the ratio of the true areally averaged rainfall to the corresponding radar-rainfall. Thus, the bias of 1.0 implies that the mean field obtained by radar is perfect. This definition allows for local discrepancies of the two. The bias is approximated replacing the true rainfall with raingage observed rainfall. This introduces sampling error due to sparseness of the raingage network.
The temporal evolution of the bias process is modeled as either a random walk process, or as a more general AR(1) process. Both models are characterized by a modeling error, the model error variance is in general unknown but in principle it can be estimated in the model calibration process. The updating of the mean field bias is achieved in all the models using the Kalman filter approach, which optimally combines the model prediction and the new observations. The Kalman filter requires specification of the observational error. This step is achieved in different ways in these models. The Seo et al. (1995) algorithm, which is a modified version of that proposed by Hudlow et al. (1983), uses radar-based spatial covariance function to determine the sampling error is calculation of the raingage based mean rainfall. This approach bets on the favorable trade-off between large sample covariance and the noise of the radar observations. Lin and Krajewski (1991) use the same bias model but propose to estimate the model and observation error variance in an adaptive way based an a well-known algorithm of Mehra (1970). The algorithm enforces lack of correlation in the innovation sequence yielding optimal performance of the filter. Smith and Krajewski (1991) developed and tested through simulation a parameter estimation scheme based on a maximum likelihood approach. It performs very well but requires a rather long data sample to converge to optimal values.
All these approaches are being tested now using operationally available data from the WSR-88D radars in the United States. Preliminary studies indicate ability of these methods to reduce hourly rainfall bias to below 20% from as high as 60% for unadjusted fields. For some applications it might be more prudent to make the mean field adjustment on the level of 3-hourly or 6-hourly accumulations. For example, the routine flow forecasting in the United States is done on 6-hourly basis and the input required is mean areal rainfall accumulated over such period. Further improvement of radar-rainfall analysis can be expected by local adjustment of the radar field using rain gages. Principles of a statistical approach to this problem are described next.
The motivation for local adjustments of radar-rainfall using rain gages lies in the fact that both sensors have the ability to estimate the same quantity. Their error structure is quite different, however, and in many respects complementary to each other. Many authors discussed both deterministic and statistical approaches to the problem. These include Brandes (1975), Crawford (1979), Ahnert et al. (1983), Krajewski (1987), Seo et al. (1990a,b), Azimi-Zonooz et al. (1989), Smith and Seo (1992), and Ciach et al. (1995). In this paper we focus on operational aspects of implementing such adjustments.
Statistical merging of radar and raingage estimates of rainfall is based on the principles of optimal estimation theory. If uncertainty characteristics of both sensors are known, a minimum variance estimator can be constructed. Advantage is taken of the fact that rainfall displays a certain degree of spatial correlation. Therefore, raingage observations, though typically sparse, exert some influence, in terms of correlation, on locations some distance away. The distance depends on the spatial covariance function of the rainfall process at the scale of interest. For rainfall averaged over larger spatial scales and integrated over longer periods, this correlation distance is typically larger.
A critical element of the statistical merging of radar and raingage sensors is the estimation of spatial covariance function. This function can be estimated either based on raingage data or radar data. There are trade-offs involved. Radar observations provide sample large enough to calculate the covariance function without much effect of the statistical sampling error, on the other hand, the radar estimates may be corrupted with noise having a structure that could obscure the true rainfall statistics. Rain gages are often too few to be useful unless a climatological ensemble approach is used (Ciach et al., 1995).
Merging is effective if the density of the raingage network is compatible with the correlation distance of the rainfall process. If the typical inter gage spacing is larger than the correlation distance only limited areas will be affected by the adjustment. If, on the other hand, the spacing is small with respect to rainfall correlation distance fewer gages may suffice for effective correction. The correction should be done in such a way as to not affect the rainfall field delineation. As radar can detect rainfall much better than a raingage network, the final rain field pattern should be that determined by radar. Raingage observations serve to make local adjustments of the estimated rainfall field. The amount of the adjustment depends on the difference between the two sensors, the location with respect to the rain gages, and the correlation function. For grid locations far from any raingage virtually no adjustment will be made.
For the adjustment to be effective it should be performed on bias-adjusted fields. Also, the uncertainty associated with the sensors involved should be specified. For the gages, this uncertainty is the result of the sampling and point measurement error and for time scales on the order of one day it can be determined based on the spatial correlation function. For shorter time scales the small scale variability of rainfall, which is not very well recognized, makes things more difficult. The most difficult task is, however, determination of radar associated uncertainty. The best approach seems to be calibration of its parameterization (Ciach et al., 1995) using radar and raingage data at time and space scales where the variability of rainfall and the random measurement noise are sufficiently smoothed out.
In statistical interpolation techniques, such as kriging, it is a common practice to include only a number of local points in estimation at a particular location. This practice may lead to artifacts in radar-rainfall field and should be avoided. An alternative solution is to include all the gages for all locations. The minimal effect of those too far will be automatically determined if minimum variance techniques are used but the resultant field will look smooth and artifact free.
One may be concerned that the above approach of statistical merging may result in excessive smoothing of the estimated field hiding important details of its pattern. It is a valid concern. On the other hand, the statistical merging discussed herein guarantees minimum error variance estimates. The details often seen in radar images are often artifact of random noise. Since it is difficult, if not impossible, to determine their realism, for applications where quantitative rainfall input is important together with its accuracy description, the merging approach is recommended.
A quite different, but potentially the most promising approach to the problem of rainfall estimation and forecasting is physically-based merging of observation from many sensors as long as they are relevant to rainfall processes. This approach of combining mathematical models of the physical processes and the operationally available observations is also referred to as data assimilation.
Limited scope of this paper prevents extensive discussion of the merits of this approach. The principles of using it include: (1) model building; (2) specification of the relationship of the model states and the available observations; (3) specification of the model and observational uncertainties; and (4) development of an updating scheme that combines the model and the observed quantities in an optimal way according to some specific performance criterion. For example, the second principle implies that the conversion of the radar-observed reflectivity can be related to rainfall rate through a parameterization of the drop size distribution rather than an empirical Z-R relationship.
This approach has been demonstrated by Lee and Georgakakos (1990), Georgakakos and Krajewski (1991), Seo and Smith (1993), French and Krajewski (1993), French et al. (1993), and Andrieu et al. (1995).
In the context of the following discussion validation is understood as the quantitative evaluation of radar rainfall estimates by comparison with independently obtained rainfall estimates. These independent estimates are commonly based on raingage observations. Several principles should be involved in validation. First, the reference estimate should be obtained by means independent of the estimate being evaluated. Second, the reference estimate should be characterized with respect to its accuracy. Third, the performance criteria of the evaluation scheme should be specified. In the following, we discuss sources of differences that may arise between the reference and the evaluated quantities.
Observed differences between the reference and evaluated quantities arise due to a number of causes, not all being sources of error. For example, if a single raingage observation is compared with radar-rainfall estimate, it is plausible, though unlikely, that both are perfect (with no error) but different from each other. The observed difference is obviously a manifestation of the sampling differences between the two sensors. To mitigate the effects of the sampling differences, validation should be performed using matched resolution, in both time and space, of the reference and the evaluated estimates. In our example, this would lead to introduction of sampling error to the estimate based on the raingage. This is a general characteristic of validation process. It involves sampling and interpolation errors due to the fact that no sensor is capable of measuring rainfall at time and space scales of interest in most engineering and scientific applications.
Interpolation of the reference and evaluated quantities to a common space-time grid requires understanding of the following aspects of radar-rainfall measurement. First, rainfall is characterized by highly intermittent behavior. Rain/no-rain intermittence is highly relevant to the validation problem as radar can "see" a much larger atmospheric space than the rain gages, and thus, it distorts the view of the true intermittence. Raingage-based interpolated rainfall fields are even more prone to excessive smoothing. This problem is more pronounced at short time scales as on longer time scales the rainfall process does its own smoothing by simple averaging mechanism. Therefore, a useful question asked in validating radar-rainfall might be: What is the time scale necessary to achieve a high probability that a given spatial scale will be fully covered by non-zero rain? The answer is complex and depends on the intermittence structure of instantaneous rainfall rate, size of the area, conditional (on non-zero rain) covariance structure of rainfall, and the frequency of storms. Preliminary results (G. Ciach, personal communication) using high (300 m) resolution radar data and a raingage network indicate the this time scale may be as high as 3-5 days!
Another problem that must be dealt with is the high degree of conditional variability of rainfall. Conditioning is based here on the fact that it rains everywhere in a given area. Again, the variability increases as the time scale decreases. As we discussed before, radar accuracy should improve with time integration. We mentioned empirical correlation between radar-rainfall and raingage rainfall at hourly scale being about 0.7. Still, absolute statements about accuracy of radar-based hourly rainfall is made difficult due to the discussed high degree of variability of rainfall at this time scale and over very short distances. As partial evidence of the variability consider hourly rainfall data obtained by two sets of rain gages in Poland. The two sets were installed at six locations within an experimental basin of the Wilga River south-east of Warsaw, Poland. The separation distances ranged from 3 to 40 meters. The differences recorded by the two sets of gages are shown in Figure 3. There are only two potential sources of the observed differences: small scale natural variability of rainfall and the raingage measurement error. Not much is known about either of them.
Fig. 3 - Comparison of hourly accumulations for collocated digital and analog raingages. Wilga River basin in Poland, June-August, 1994. Distance between raingages are given below the station names.
Measurement errors in raingage observations of rainfall are well-acknowledged (Sevruk, 1985). The strongest effects leading to systematic errors (undercatch or underestimation) are caused by bad exposure and strong wind. Several researchers recommended correction to be introduced to raingage observations. Such corrections, introduced on monthly or annual scales and critical for water balance studies, range from 2-10% depending on climatic conditions and instrument type. Still, for validation of radar-rainfall at short time scales random errors need to be better understood. As experimental studies are difficult for technical reasons, the most promising approach seems to be numerical calculations based on methods of computational fluid dynamics (Nespor, 1995).
A critical element of validation studies is selection of performance criteria. It seems that the most plausible approach is that based on probabilistic description of the errors. If errors are normally distributed, or if they can be easily transformed to such, the mean errors and the error estimation variance provide full characterization. This error characterization should be applied to the final products of interest. These final products may be, for example, daily rainfall accumulations grided maps. The quality of the final products should also drive the calibration process. Consider, as an example, sensitivity of the daily rainfall accumulations obtained at 2¥2 km resolution over a 25 day period in Darwin, Australia. Figure 4 shows contour plots of the root mean square differences between the radar estimates and the daily raingage observations in parameter space of Z-R relationship. The plot clearly shows that there is not much sensitivity, over most of the parameter domain, with respect to the exponent of the relationship with the scaling parameter adjusted to match the total period accumulation. For details of the presented results and the applied estimation algorithm refer to Ciach et al. (1995).
Fig. 4 - Contour plot of the mean square error of daily rainfall accumulations in a two-dimensional domain of Z-R relationship parameters. Prior to estimation rainfall classification was performed and different Z-R parameters were applied to convective and stratiform rainfall. Ac/As is the ratio of the convective to stratiform Z-R multiplier. The exponent B was common for both regimes.
The above example clearly points towards the fact that successful radar-rainfall estimation involves much more than Z-R specification. It also emphasizes the need for calibration driven by the quality of the final products. For the off-line applications, such as climatic variability studies, or satellite rainfall validation, the calibration could be viewed as a multidimensional optimization process. The decision variables are the parameters of the estimation algorithm. These include the Z-R relationship parameters, the reflectivity threshold, the CAPPI level, range effects parameterization, etc. The real-time (on-line) applications should incorporate a scheme for on-line parameter estimation or periodic batch reprocessing of the data.
Both validation and calibration should be performed on large samples. One or two storm case studies are not adequate due to the tremendous variability of the atmospheric conditions associated with rainfall that affect the measurement and estimation process. The above needs call for efficient data storage and database organization (Kruger and Krajewski, 1995).
We presented a brief discussion of the use of raingages in radar-rainfall estimation. The main conclusion is that both radar and raingage networks are equally important in most applications. Radar, being an indirect sensor of rainfall, needs calibration that can be provided by use of raingages. The rain gages, on the other hand, are typically too sparse and logistically too complex to detect many details of storms important for real time hydrologic applications and climatological studies of rainfall and its extremes.
The most pressing research needs are in characterization of rainfall variability, especially at small scale.The variability at spatial scale smaller than the resolution of the radar sampling volume might have important implications for understanding of the averaging properties of radar. Rainfall intermittence and its scaling properties (Foufoula-Georgiou and Krajewski, 1995) affect the transformation of point to areally averaged rainfall. Studies on uncertainty of various measurement and estimation methods, both involving rain gages and radars, are critical for validation and accuracy determination. The uncertainty sources that need to be better understood include measurement error, sampling error, and the estimation error associated with various algorithms. The methods involved in error studies might involve careful inter sensor comparisons, simulation studies, and error propagation using analytical and numerical methods.
This work was supported by NASA grant NAG 5-2084, by the United States Agency for International Development under Grant HRN-5600-G-00-2037-00, and by NOAA under Cooperative Agreement between the Office of Hydrology of the National Weather Service and the Iowa Institute of Hydraulic Research. The author acknowledges helpful discussions with and/or material contribution from Grzegorz Ciach, James Smith, Emmanouil Anagnostou, Stanislaw Moszkowicz, and Jeffrey McCollum.
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